Geometry

In this episode of the I Can't Sleep Podcast, fall asleep learning about Geometry. Shapes, objects, lines, points...wow! That took a lot out of me just typing those words. I think I'll go lie down for a nap. Happy sleeping!

This content is derived from the Wikipedia article Geometry, available under the Creative Commons Attribution-ShareAlike (CC BY-SA) license.

Transcript

Welcome to the I Can't Sleep Podcast,

Where I read random articles from across the web to bore you to sleep with my soothing voice.

I'm your host,

Benjamin Boster.

Today's episode is from a Wikipedia article titled,

Geometry.

Geometry from Ancient Greek Geometria,

Land Measurement,

From Earth,

Land,

And Metron,

A measure,

Is a branch of mathematics concerned with properties of space,

Such as the distance,

Shape,

Size,

And relative position of figures.

Geometry is,

Along with arithmetic,

One of the oldest branches of mathematics.

A mathematician who works in the field of geometry is called a geometer.

Until the 19th century,

Geometry was almost exclusively devoted to Euclidean geometry,

Which includes the notions of point,

Line,

Plane,

Distance,

Angle,

Surface,

And curve as fundamental concepts.

Originally developed to model the physical world,

Geometry has applications in almost all sciences,

And also in art,

Architecture,

And other activities that are related to graphics.

Geometry also has applications in areas of mathematics that are apparently unrelated.

For example,

Methods of algebraic geometry are fundamental in Wiles' proof of Fermat's Last Theorem,

A problem that was stated in terms of elementary arithmetic,

And remained unsolved for several centuries.

During the 19th century,

Several discoveries enlarged dramatically the scope of geometry.

One of the oldest such discoveries is Carl Friedrich Gauss's Theorema Egregium,

Remarkable Theorem that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in Euclidean space.

This implies that surfaces can be studied intrinsically,

That is,

As stand-alone spaces,

And has been expanded into the theory of manifolds and Riemannian geometry.

Later in the 19th century,

It appeared that geometries without the parallel postulate non-Euclidean geometries can be developed without introducing any contradiction.

The geometry that underlies general relativity is a famous application of non-Euclidean geometry.

Since the late 19th century,

The scope of geometry has been greatly expanded,

And the field has been split in many sub-fields that depend on the underlying methods.

Differential Geometry,

Algebraic Geometry,

Computational Geometry,

Algebraic Topology,

Discrete Geometry,

Also known as Combinatorial Geometry,

Etc.

Or on the properties of Euclidean spaces that are disregarded,

Projective geometry that consider only alignment of points,

But not distance and parallelism,

Affine geometry that omits the concept of angle and distance,

Finite geometry that omits continuity,

And others.

This enlargement of the scope of geometry led to a change of meaning of the word space,

Which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry.

Presently,

A geometric space,

Or simply a space,

Is a mathematical structure on which some geometry is defined.

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.

Early geometry was the collection of empirically discovered principles concerning lengths,

Angles,

Areas,

And volumes,

Which were developed to meet some practical need in surveying,

Construction,

Astronomy,

And various crafts.

The earliest known texts on geometry are the Egyptian rind papyrus and Moscow papyrus and the Babylonian clay tablets,

Such as Plimpton 322.

For example,

The Moscow papyrus gives a formula for calculating the volume of a truncated pyramid,

Or frustum.

Later,

Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space.

These geometric procedures anticipated the Oxford calculators,

Including the mean speed theorem,

By fourteen centuries.

Across Egypt,

The ancient Nubians established a system of geometry,

Including early versions of sun clocks.

In the 7th century BC,

The Greek mathematician Thales of Miletus used geometry to solve problems,

Such as calculating the height of pyramids and the distance of ships from the shore.

He is credited with the first use of deductive reasoning applied to geometry by deriving four corollaries to Thales' theorem.

Pythagoras established the Pythagorean school,

Which is credited with the first proof of the Pythagorean theorem,

Though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion,

Which allowed the calculation of areas and volumes of curvilinear figures,

As well as a theory of ratios that avoided the problem of incommeasurable magnitudes,

Which enabled subsequent geometers to make significant advances.

Around 300 BC,

Geometry was revolutionized by Euclid,

Whose Elements,

Widely considered the most successful and influential textbook of all time,

Introduced mathematical rigor through the axiomatic method,

And is the earliest example of the format still used in mathematics today,

That of definition,

Axiom,

Theorem,

And proof.

Although most of the contents of the Elements were already known,

Euclid arranged them into a single coherent logical framework.

The Elements was known to all educated people in the West until the middle of the 20th century,

And its contents are still taught in geometry classes today.

Archimedes of Syracuse,

Italy,

Used the method of exhaustion to calculate the area under the arc of a parabola,

With the summation of an infinite series,

And gave remarkably accurate approximations of pi.

He also studied the spiral bearing,

His name,

And obtained formulas for the volumes of surfaces of revolution.

Indian mathematicians also made many important contributions in geometry.

The Shatabatha Brahman contains rules for its ritual geometric constructions that are similar to the Sulba Sutras.

According to Hayashi,

2005,

Page 363,

The Sulba Sutras contain the earliest extant verbal expression of the Pythagorean theorem in the world,

Although it had already been known to the old Babylonians.

They contain lists of Pythagorean triples,

Which are particular cases of Diophantine equations.

In the Bhakshali Manuscript,

There are a handful of geometric problems,

Including problems about volumes of irregular solids.

The Bhakshali Manuscript also employs a decimal place value system with a dot for zero.

Aryabhata's Aryabhatiya includes the computation of areas and volumes.

Brahmagupta wrote his astronomical work in 628.

Chapter 12,

Containing 66 Sanskrit verses,

Was divided into two sections,

Basic Operations,

Including cube roots,

Fractions,

Ratio and proportion,

And barter,

And Practical Mathematics,

Including mixture,

Mathematical series,

Plane figures,

Stacking bricks,

Sawing of timber,

And piling of grain.

In the latter section,

He stated his famous theorem on the diagonals of a cyclical quadrilateral.

Chapter 12 also included a formula for the area of a cyclic quadrilateral,

A generalization of Heron's formula,

As well as a complete description of rational triangles,

I.

E.

Triangles with rational sides and rational areas.

In the Middle Ages,

Mathematics in medieval Islam contributed to the development of geometry,

Especially algebraic geometry.

Al-Mahhani conceived the idea of reducing geometrical problems,

Such as duplicating the cube,

To problems in algebra.

Thabit ibn Khura,

Known as Thabit in Latin,

Dealt with arithmetic operations applied to ratios of geometrical quantities,

And contributed to the development of analytic geometry.

Omar Khayyam found geometric solutions to cubic equations.

The theorems of Ibn al-Haytham,

Omar Khayyam,

And Nasir al-Din al-Tusi on quadrilaterals,

Including the Lambert quadrilateral and Saatchiri quadrilateral,

Were early results in hyperbolic geometry,

And along with their alternative postulates,

Such as Playfair's axiom.

These works had a considerable influence on the development of non-Euclidean geometry among later European geometers,

Including Vitello,

Gersonides,

Alfonso,

John Wallace,

And Giovanni Giorlamo Saatchiri.

In the early 17th century,

There were two important developments in geometry.

The first was the creation of analytic geometry,

Or geometry with coordinates and equations,

By René Descartes and Pierre de Fermat.

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

The second geometric development of this period was the systematic study of projective geometry by Gérard Désargues.

Projective geometry studies properties of shapes which are unchanged under projections and sections,

Especially as they relate to artistic perspective.

Two developments in geometry in the 19th century changed the way it had been studied previously.

These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky,

Janos Bulyai,

And Carl Friedrich Gauss,

And of the formulation of symmetry as a central consideration in the Erlangen program of Felix Klein,

Which generalized the Euclidean and non-Euclidean geometries.

Two of the master geometers of the time were Bernard Riemann,

Working primarily with tools for mathematical analysis and introducing the Riemann surface,

And Henri Poincaré,

The founder of algebraic topology and the geometric theory of dynamical systems.

As a consequence of these major changes in the conception of geometry,

The concept of space became something rich and varied,

And the natural background for theories as different as complex analysis and classical mechanics.

The following are some of the most important concepts in geometry.

Axioms.

Euclid took an abstract approach to geometry and his elements,

One of the most influential books ever written.

Euclid introduced certain axioms or postulates,

Expressing primary or self-evident properties of points,

Lines,

And planes.

He proceeded to rigorously deduce other properties by mathematical reasoning.

The characteristic feature of Euclid's approach to geometry was its rigor,

And it has come to be known as axiomatic or synthetic geometry.

At the start of the 19th century,

The discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky,

Janos Poliai,

And Carl Friedrich Gauss,

And others,

Led to a revival of interest in this discipline,

And in the 20th century,

David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.

Objects.

Points.

Objects are generally considered fundamental objects for building geometry.

They may be defined by the properties that they must have,

As in Euclid's definition as that which has no part,

Or in synthetic geometry.

In modern mathematics,

They are generally defined as elements of a set called space,

Which is itself axiomatically defined.

With these modern definitions,

Every geometric shape is defined as a set of points.

This is not the case in synthetic geometry,

Where a line is another fundamental object that is not viewed as the set of the points through which it passes.

However,

There are modern geometries in which points are not primitive objects,

Or even without points.

One of the oldest such geometries is Whitehead's point-free geometry,

Formulated by Alfred North Whitehead in 1919-1920.

Lines.

Euclid described a line as breathless length,

Which lies equally with respect to the points on itself.

In modern mathematics,

Given the multitude of geometries,

The concept of a line is closely tied to the way the geometry is described.

For instance,

In analytic geometry,

A line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation.

But in a more abstract setting,

Such as incidence geometry,

A line may be an independent object,

Distinct from the set of points which lie on it.

In differential geometry,

A geodesic is a generalization of the notion of a line to curved spaces.

In Euclidean geometry,

A plane is a flat two-dimensional surface that extends infinitely.

The definition for other types of geometries are generalizations of that.

Planes are used in many areas of geometry.

For instance,

Planes can be studied as a tropological surface,

Without reference to distances or angles.

It can be studied as an affine space where collinearity and ratios can be studied,

But not distances.

It can be studied as the complex plane using techniques of complex analysis,

And so on.

Angles Euclid defines a plane angle as the inclination to each other in a plane of two lines which meet each other,

And do not lie straight with respect to each other.

In modern terms,

An angle is the figure formed by two rays,

Called the sides of the angle,

Sharing a common endpoint,

Called the vertex of the angle.

In Euclidean geometry,

Angles are used to study polygons and triangles,

As well as forming an object of study in their own right.

The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.

In differential geometry and calculus,

The angles between plane curves or space curves or surfaces can be calculated using the derivative.

Curves A curve is a one-dimensional object that may be straight,

Like a line,

Or not.

Curves in two-dimensional space are called plane curves,

And those in three-dimensional space are called space curves.

In topology,

A curve is defined by a function from an interval of the real numbers to another space.

In differential geometry,

The same definition is used,

But the defining function is required to be differentiable.

Algebraic geometry studies algebraic curves,

Which are defined as algebraic varieties of dimension one.

Surfaces A surface is a two-dimensional object,

Such as a sphere or paraboloid.

In differential geometry and topology,

Surfaces are described by two-dimensional patches or neighborhoods that are assembled by diffeomorphisms or homeomorphisms,

Respectively.

In algebraic geometry,

Surfaces are described by polynomial equations.

Solids A solid is a three-dimensional object bounded by a closed surface.

For example,

A ball is the volume bounded by a sphere.

Manifolds A manifold is a generalization of the concepts of curve and surface.

In topology,

A manifold is a topological space,

Where every point has a neighborhood that is homeomorphic to Euclidean space.

In differential geometry,

A differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.

Manifolds are used extensively in physics,

Including in general relativity and string theory.

Measures Length,

Area,

And volume Length,

Area,

And volume describe the size or extent of an object in one dimension,

Two dimension,

And three dimensions,

Respectively.

In Euclidean geometry and analytic geometry,

The length of a line segment can often be calculated by the Pythagorean theorem.

Area and volume can be defined as fundamental quantities separated from length,

Or they can be described and calculated in terms of lengths in a plane or three-dimensional space.

Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects.

In calculus,

Area and volume can be defined in terms of integrals,

Such as the Riemann integral or the Lebesgue integral.

Other geometrical measures include the angular measure,

Curvature,

Compactness measures.

The concept of length or distance can be generalized,

Leading to the idea of metrics.

For instance,

The Euclidean metric measures the distance between points in the Euclidean plane,

While the hyperbolic metric measures the distance in the hyperbolic plane.

Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.

In a different direction,

The concepts of length,

Area,

And volume are extended by measure theory,

Which studies methods of assigning a size or measure to sets,

Where the measures follow rules similar to those of classical area and volume.

Congruence and similarity are concepts that describe when two shapes have similar characteristics.

In Euclidean geometry,

Similarity is used to describe objects that have the same shape,

While congruence is used to describe objects that are the same in both size and shape.

Hilbert,

In his work on creating a more rigorous foundation for geometry,

Treated congruence as an undefined term whose properties are defined by axioms.

Congruence and similarity are generalized in transformation geometry,

Which studies the properties of geometric objects that are preserved by different kinds of transformations.

Classical geometers paid special attention to constructing geometric objects that had been described in some other way.

Classically,

The only instruments used in most geometric constructions are the compass and straightedge.

Also,

Every construction had to be complete in a finite number of steps.

However,

Some problems turned out to be difficult or impossible to solve by these means alone.

Even ingenious constructions using gnosis,

Parabolas,

And other curves or mechanical devices were found.

The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.

Where the traditional geometry allowed dimensions 1,

A line,

2,

A plane,

And 3,

Our ambient world conceived of as three-dimensional space,

Mathematicians and physicists have used higher dimensions for nearly two centuries.

One example of a mathematical use for higher dimensions is the configuration space of a physical system,

Which has a dimension equal to the system's degrees of freedom.

For instance,

The configuration of a screw can be described by five coordinates.

In general topology,

The concept of dimension has been extended from natural numbers to infinite dimension and positive real numbers.

In algebraic geometry,

The dimension of an algebraic variety has received a number of apparently different definitions,

Which are all equivalent in the most common cases.

The theme of symmetry in geometry is nearly as old as the science of geometry itself.

Symmetry shapes such as the circle,

Regular polygons,

And platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid.

Symmetric patterns occur in nature and were artistically rendered in a multitude of forms,

Including the graphics of Leonardo da Vinci,

M.

C.

Escher,

And others.

In the second half of the 19th century,

The relationship between symmetry and geometry came under intense scrutiny.

Felix Klein's Erlangen program proclaimed that,

In a very precise sense,

Symmetry expressed via the notion of a transformation group determines what geometry is.

Symmetry in classical Euclidean geometry is represented by congruences and rigid motions,

Whereas in projective geometry an analogous role is played by collineations,

Geometric transformations that take straight lines into straight lines.

However,

It was in the new geometries of Bolyai and Lobachevsky,

Ryman,

Clifford,

And Klein,

And Sophus-Lee,

That Klein's idea to define a geometry via its symmetry group found its inspiration.

Both discrete and continuous symmetries play prominent roles in geometry,

The former in topology and geometric group theory,

The latter in Lie theory and Ramanian geometry.

A different type of symmetry is the principle of duality in projective geometry,

Among other fields.

This metaphenomenon can roughly be described as follows.

In any theorem,

Exchange point with plane,

Join with meet,

Lies in with contains,

And the result is an equally true theorem.

A similar and closely related form of duality exists between a vector space and its dual space.

Euclidean geometry is geometry in its classical sense,

As it models the space of the physical world.

As it models the space of the physical world,

It is used in many scientific areas,

Such as mechanics,

Astronomy,

Crystallography,

And many technical fields such as engineering,

Architecture,

Geodesy,

Aerodynamics,

And navigation.

The mandatory educational curriculum of the majority of nations includes a study of Euclidean concepts such as points,

Lines,

Planes,

Angles,

Triangles,

Congruence,

Similarity,

Solid figures,

Circles,

And analytic geometry.

Euclidean vectors are used for a myriad of applications in physics and engineering,

Such as position,

Displacement,

Deformation,

Velocity,

Acceleration,

Force,

Etc.

Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.

It has applications in physics,

Econometrics,

And bioinformatics,

Among others.

In particular,

Differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that the universe is curved.

Differential geometry can either be intrinsic,

Meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Ramanian metric,

Which determines how distances are measured near each point,

Or extrinsic,

Where the object under study is a part of some ambient flat Euclidean space.

In mathematics,

Non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.

As Euclidean geometry lies at the intersection of metric geometry and defined geometry,

Non-Euclidean geometry arises by either replacing the parallel postulate with an alternative,

Or relaxing the metric requirement.

In the former case,

One obtains hyperbolic geometry and elliptic geometry,

The traditional non-Euclidean geometries.

When the metric requirement is relaxed,

Then there are affine planes associated with the planar algebras,

Which give rise to kinematic geometries that have also been called non-Euclidean geometry.

Topology is the field concerned with the properties of continuous mappings,

And can be considered a generalization of Euclidean geometry.

In practice,

Topology often means dealing with large-scale properties of spaces,

Such as connectedness and compactness.

The field of topology,

Which saw massive development in the 20th century,

Is in a technical sense a type of transformation geometry,

In which transformations are homeomorphisms.

This has often been expressed in the form of the saying,

Topology is rubber-sheet geometry.

Subfields of topology include geometric topology,

Differential topology,

Algebraic topology,

And general topology.

Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shape,

Called algebraic sets,

And defined as common zeros of multivariate polynomials.

Algebraic geometry became an autonomous subfield of geometry around 1900,

With a theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings.

This led to a parallel development of algebraic geometry,

And its algebraic counterpart called commutative algebra.

From the 1950s through the mid-1970s,

Algebraic geometry had undergone major foundational development with the introduction by Alexander Groszendieck of scheme theory,

Which allows using topological methods,

Including cohomology theories in a purely algebraic context.

Scheme theory allowed to solve many difficult problems not only in geometry,

But also in number theory.

Wiles' proof of Fermat's last theorem is a famous example of a long-standing problem of number theory,

Whose solution uses scheme theory and its extensions such as stack theory.

One of seven Millennium Prize problems,

The Hodge conjecture,

Is a question in algebraic geometry.

Algebraic geometry has applications in many areas,

Including cryptography and string theory.

Complex geometry studies the nature of geometric structures modeled on,

Or arising out of,

The complex plane.

Complex geometry lies at the intersection of differential geometry,

Algebraic geometry,

And analysis of several complex variables,

And has found applications to string theory and mirror symmetry.

Complex geometry first appeared as a distinct area of study in the work of Bernard Riemann and his study of Riemann surfaces.

Work in the spirit of Riemann was carried out by the Italian School of Algebraic Geometry in the early 1900s.

Contemporary treatment of complex geometry began with the work of Jean-Pierre Serres,

Who introduced the concept of sheaves to the subject,

And illuminated the relations between complex geometry and algebraic geometry.

The primary objects of study in complex geometry are complex manifolds,

Complex algebraic varieties,

And complex analytic varieties,

And holomorphic vector bundles,

And coherent sheaves over these spaces.

Special examples of spaces studied in complex geometry include Riemann surfaces and Calabi-Yau manifolds,

And these spaces find uses in string theory.

In particular,

World sheets of strings are modeled by Riemann surfaces,

And superstring theory predicts that the extra six dimensions of ten-dimensional spacetime may be modeled by Calabi-Yau manifolds.

Discrete geometry is a subject that has close connections with convex geometry.

It is concerned mainly with questions of relative position of simple geometric objects,

Such as points,

Lines,

And circles.

Examples include the study of sphere packings,

Triangulation,

The Nieser-Polson conjecture,

Etc.

It shares many methods and principles with combinatorics.

Computational geometry deals with algorithms and their implementations for manipulating geometrical objects.

Important problems historically have included the traveling salesman problem,

Minimum spanning trees,

Hidden line removal,

And linear programming.

Although being a young area of geometry,

It has many applications in computer vision,

Image processing,

Computer-aided design,

Medical imaging,

Etc.

Geometric group theory uses large-scale geometric techniques to study finitely generated groups.

It is closely connected to low-dimensional topology,

Such as Ingrigore Perelman's proof of the geometrization conjecture,

Which included the proof of the Poincaré conjecture,

A millennium prize problem.

Geometric group theory often revolves around the Cayley graph,

Which is a geometric representation of a group.

Other important topics include quasi-isometries,

Gromov hyperbolic groups,

And right-angled Ardennes groups.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues,

Often using techniques of real analysis and discrete mathematics.

It has close connections to convex analysis,

Optimization and functional analysis,

And important applications in number theory.

Convex geometry dates back to antiquity.

Archimedes gave the first known precise definition of convexity.

The isoparametric problem,

A recurring concept in convex geometry,

Was studied by the Greeks as well,

Including Xenodorus.

Archimedes,

Plato,

Euclid,

And later Kepler and Coxeter all studied convex polytopes and their properties.

From the 19th century on,

Mathematicians have studied other areas of complex mathematics,

Including higher-dimensional polytopes,

Volume and surface areas of convex bodies,

Gaussian curvature,

Algorithms,

Tilings,

And lattices.

Geometry has found applications in many fields.

Art Mathematics and art are related in a variety of ways.

For instance,

The theory of perspective showed that there is more to geometry than just the metric properties of figures.

Perspective is the origin of projective geometry.

Artists have long used concepts of proportion in design.

Vitruvius developed a complicated theory of ideal proportions for the human figure.

These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.

The golden ratio is a particular proportion that has had a controversial role in art.

Often claimed to be the most aesthetically pleasing ratio of length,

It is frequently stated to be incorporated into famous works of art,

Though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.

Tilings or tessellations have been used in art throughout history.

Islamic art makes frequent use of tessellations,

As did the art of M.

C.

Escher.

Escher's work also made use of hyperbolic geometry.

Cézanne advanced the theory that all images can be built up from the sphere,

The cone,

And the cylinder.

This is still used in art theory today,

Although the exact list of shapes varies from author to author.

Architecture Geometry has many applications in architecture.

In fact,

It has been said that geometry lies at the core of architectural design.

Applications of geometry to architecture include the use of projective geometry to create forced perspective,

The use of conic sections in construction domes and similar objects,

The use of tessellations,

And the use of symmetry.

Physics The field of astronomy,

Especially as it relates to mapping the positions of stars and planets on the celestial sphere,

And describing the relationship between movements of celestial bodies,

Have served as an important source of geometric problems throughout history.

Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.

String theory makes use of several variants of geometry,

As does quantum information theory.

Other fields of mathematics Calculus was strongly influenced by geometry.

For instance,

The introduction of coordinates by Rene Descartes and the concurrent developments of algebra marked a new stage for geometry,

Since geometric figures such as plane curves could now be represented analytically in the form of functions and equations.

This played a key role in the emergence of infinitesimal calculus in the 17th century.

Analytic geometry continues to be a mainstay of precalculus and calculus curriculum.

Another important area of application is number theory.

In ancient Greece,

The Pythagoreans considered the role of numbers in geometry.

However,

The discovery of incommensurable lengths contradicted their philosophical views.

Since the 19th century,

Geometry has been used for solving problems in number theory.

For example,

Through the geometry of numbers,

Or more recently,

Scheme theory,

Which is used in Weyl's proof of Fermat's last theorem.

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